Question

$\cos (2x+7)=a(2-\sin x)$ can have a real solution for

• A. all real values of $a$
• B. $a\in [2,6]$
• C. $a\in (-\infty ,2)\setminus \left\{0\right\}$
• D. $a\in (0,\infty )$

Answer 1

The correct option is B $a\in [2,6]$

Note: There is an error in this question.
The correct equation is $\cos 2x+7=a(2-\sin x)$

$\cos 2x+7=a(2-\sin x)$
$\Rightarrow 1-2{\sin }^{2}x+7=2a-a\sin x$
$\Rightarrow 2{\sin }^{2}x-a\sin x+2a-8$
$\Rightarrow \sin x=\frac{a\pm \sqrt{a^2-8(2a-8)}}{4}\Rightarrow \sin x=\frac{a\pm (a-8)}{4}\Rightarrow \sin x=\frac{a-4}{2}$

We know that,
$-1\le \sin x\le 1\Rightarrow -1\le \frac{a-4}{2}\le 1\Rightarrow 2\le a\le 6$